Weight painting on low poly model acts weird
You have duplicate vertices in your model, and you are working in opaque mode ('limit selection to visible'), so only one of the duplicates can be selected. Several ways to get around this, one would be to select the vertives you want to work on in edit mode, then switch to weight paint and activate the icon directly to the left of the layer selection block (a cube with one highlighted vertex, "vertex selection masking"). Then you'll be able to only work on the selected vertices (you might need to unselect 'limit selection to visible').

Or you could remove the doubles in edit mode...